Most important Linear Algebra theorems?

Question

I was reading up on symmetric matrices and the textbook noted that the following is a remarkable theorem:

A matrix $A$ is orthogonally diagonalizable iff $A$ is a symmetric matrix.

This is because it is impossible to tell when a matrix is diagonalizable, or so it seems.

I haven't gotten to realize yet how important this is, but I will soon. What, in your opinion , is the most important linear algebra theorem and why?

It is a nice theorem which tells you a lot of facts in one shot. — Mhenni Benghorbal, Dec 02 '13 at 15:37

score 4 · Answer 1 · answered Dec 02 '13 at 15:40

4

The two main candidates are:

The fundamental theorem of linear algebra, as popularised by Strang.
The singular value decomposition.

From these, lots of important results follow.

answered Dec 02 '13 at 15:40

lhf

216,483

3

I think "popularised" by Strang is an understatement. I think without Strang nobody would have noticed (or cared for such a formulation). – Marc van Leeuwen Dec 02 '13 at 15:58
@MarcvanLeeuwen Really? Whyso? Because it had never really been conceptualized in such a way, or because it was seen as too simple? When I first learned the theorem, it seemed super obvious, but also quite conceptually useful. How had it gone so long without being made explicit? – Emily Dec 02 '13 at 16:51
@Arkamis: See Intuitive explanation of the Fundamental Theorem of Linear Algebra and the comments to that question, notably the one by Willie Wong. – Marc van Leeuwen Dec 02 '13 at 20:28
@MarcvanLeeuwen I see. So that your earlier comment here is to make the point that, in a way, Strang "dumbed down" what was already a fairly straightforward result? – Emily Dec 02 '13 at 20:56

score 4 · Answer 2 · answered Dec 02 '13 at 16:42

4

Undoubtly the Invertible Matrix Theorem in my opinion.

answered Dec 02 '13 at 16:42

imranfat

10,029

score 1 · Answer 3 · answered Jun 03 '18 at 15:08

Well since linear algebra is basically about studying linear functions in linear spaces, i would say the rank-nullity theorem. It basically limits the size of range in terms of the nullity, and vice-versa. THe nullity of a tranformation and the rank add up to the dimension of the space. I dont know if we can say which is the most important but this is a result worth understanding very well and it also has similar form in other structures.

Most important Linear Algebra theorems?

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